Problem 29 - Algebra A Combined Function

Question

Subtract. $$ \left(\frac{1}{4} z^{2}-\frac{1}{5} z\right)-\left(-\frac{3}{20} z^{2}+\frac{1}{10} z-\frac{7}{20}\right) $$

Step-by-Step Solution

Answer

\( \frac{2}{5} z^{2} - \frac{3}{10} z + \frac{7}{20} \)

Step 1: Distribute the Negative Sign

We need to change the signs of each term in the second polynomial because of the negative sign in front of the parenthesis. This gives us:\[ \left( \frac{1}{4} z^{2} - \frac{1}{5} z \right) + \left( \frac{3}{20} z^{2} - \frac{1}{10} z + \frac{7}{20} \right) \]

Step 2: Combine Like Terms (Quadratic Term)

Combine the terms with \(z^2\):\[ \frac{1}{4} z^{2} + \frac{3}{20} z^{2} \]Convert \(\frac{1}{4}\) to a common denominator of 20:\[ \frac{1}{4} = \frac{5}{20} \]Now, add:\[ \frac{5}{20} z^{2} + \frac{3}{20} z^{2} = \frac{8}{20} z^{2} \]This simplifies to:\[ \frac{2}{5} z^{2} \]

Step 3: Combine Like Terms (Linear Term)

Combine the terms with \(z\):\[ -\frac{1}{5} z - \frac{1}{10} z \]Convert \(\frac{1}{5}\) to a common denominator of 10:\[ -\frac{1}{5} = -\frac{2}{10} \]Now, add:\[ -\frac{2}{10} z - \frac{1}{10} z = -\frac{3}{10} z \]

Step 4: Constant Term

There is no constant term in the first polynomial, so the constant term remains:\[ +\frac{7}{20} \]

Step 5: Final Expression

Combine all the simplified terms:\[ \frac{2}{5} z^{2} - \frac{3}{10} z + \frac{7}{20} \]

Key Concepts

Distributive Property

The distributive property is a pivotal concept in algebra, especially when dealing with polynomial operations such as addition, subtraction, and multiplication. It allows us to simplify expressions by distributing a factor across terms within parentheses. In the context of subtraction, we often encounter a scenario where a negative sign is in front of a group of terms. The key step is to distribute this negative sign to each term inside the parentheses, changing their signs appropriately. For instance, \(-(-\frac{3}{20}z^{2}+\frac{1}{10}z-\frac{7}{20})\) becomes \(+\frac{3}{20}z^{2}-\frac{1}{10}z+\frac{7}{20}\) after distributing the negative sign. This step sets up the expression for further simplification and ensures accurate combination of like terms in later steps.

Combining Like Terms

Combining like terms is an essential skill when handling algebraic expressions. Like terms are terms that share the same variable to the same power. In polynomial subtraction, once the distributive property is applied, we scan through the expression to identify and group together like terms.

• Quadratic terms like \(\frac{1}{4}z^{2}\) and \(\frac{3}{20}z^{2}\) are combined because they both include \(z^2\) as a factor.
• Linear terms such as \(-\frac{1}{5}z\) and \(-\frac{1}{10}z\) are grouped together because each term includes \(z\) as a factor.

Each group of like terms is then simplified further, often requiring a common denominator for fractions before they can be added or subtracted.

Common Denominator

Finding a common denominator is crucial when adding or subtracting fractions, particularly when working with polynomial terms. When fractions have different denominators, they cannot be directly added or subtracted. We adjust them to have a common denominator, thus enabling straightforward arithmetic operations.
For example, when combining \(\frac{1}{4}z^{2}\) with \(\frac{3}{20}z^{2}\), we convert \(\frac{1}{4}\) to \(\frac{5}{20}\) because 20 is a common multiple of the denominators 4 and 20. Similarly, for \(-\frac{1}{5}z\) and \(-\frac{1}{10}z\), \(-\frac{1}{5}\) is converted to \(-\frac{2}{10}\).
This will ensure all terms are comparable and can be easily combined, leading to simpler expressions.

Simplifying Expressions

The final step in managing polynomial subtraction is simplifying expressions, which means reducing them to their most efficient form. After like terms have been combined using a common denominator, we perform essential arithmetic operations to simplify the mathematical expression further. The process might include:

• Reducing fractions, such as turning \(\frac{8}{20}z^{2}\) into \(\frac{2}{5}z^{2}\).
• Ensuring each part of the polynomial has been simplified as far as it can be, particularly regarding fractional terms.
• Combining all terms into one seamless, simplified polynomial expression.

The expression \(\frac{2}{5}z^{2}-\frac{3}{10}z+\frac{7}{20}\) is the final result after applying these steps, making it easier to understand and work with for further mathematical applications.